OpenAI and Anthropic each said on 26 October that one of their AI models had disproved the exponential time hypothesis, a nearly 30-year-old conjecture about the limits of computation. Now the scientists who proposed the idea are helping to organise an independent review of the claims — a check that will decide whether a cornerstone assumption of computer science really falls.
What is the exponential time hypothesis?
The exponential time hypothesis (ETH), proposed by Russell Impagliazzo and Ramamohan Paturi of the University of California San Diego around 1999 and formalised with Francis Zane in 2001, holds that some problems are so hard that computers can solve them only by trying nearly every possible answer. Its canonical example is 3-SAT, a Boolean satisfiability problem at the heart of NP-completeness: ETH says 3-SAT cannot be solved in subexponential time, meaning no algorithm does meaningfully better than brute-force search as the number of variables grows.
Two labs, the same claim, the same day
OpenAI announced on 26 October that an internal model had produced three new mathematical results, including a disproof of ETH, with a Lean 4 formalisation — a machine-checked proof that certifies each logical step. Hours later, Anthropic published its own paper, titled “A 2^o(n) Time Algorithm for 3-SAT”, credited simply to “Claude (Anthropic)”. It presents a deterministic algorithm for 3-SAT — and for the harder counting problem #3-SAT — in subexponential time, directly contradicting ETH. Anthropic researcher Levent Alpoge said Claude’s internal research model found the core discovery on 16 October using about 14 million output tokens, with no human involvement in the discovery itself; Anthropic researchers then worked with Claude to improve how the result was written up. Claude’s version also disproves the Strong Exponential Time Hypothesis (SETH), a stricter claim that logically implies ETH. Both labs say they had no knowledge of the other’s work.
The creators are now checking the work
In interviews with AFP published on 31 October, Impagliazzo, Paturi and Ryan Williams of MIT — a leading researcher who had long publicly doubted ETH — said they are helping organise an independent review of the companies’ claims. Paturi said his first reaction was surprise, noting ETH “has shaped a great deal of research over the past three decades”. Williams said he felt vindicated as a prominent sceptic, but added he would have preferred the discovery to be made by a human who could explain the thought process, saying he would have liked “to have shaken the hand” of the person who disproved ETH. The reviewers stress that a company’s announcement cannot substitute for the community’s assessment, and that publicly releasing the proofs matters precisely so independent experts can examine them.
What is at stake — and what is not
If the proofs survive scrutiny, researchers may have to revisit more than 800 published papers that used ETH as a foundational assumption. But the result does not resolve P vs NP, computer science’s most famous open question: the claimed bound is subexponential, not polynomial, so it does not show that hard problems can be solved quickly in general. Nor does it make anything faster in practice — the algorithms are theoretical results about asymptotic running time, not software anyone can run today.
What remains uncertain
Neither proof has been peer-reviewed. Both are lab claims, not established results, and the independent review is still under way. Anthropic’s own paper notes that its Lean development was written by AI models and reviewed by other instances of the same models, and that at the time of writing no person was known to have checked it. The honest status today is: two AI systems produced formal proofs that appear to refute ETH, and the mathematicians who built the field are now testing whether those proofs hold.
Why it matters
Beyond the mathematics, the episode is a test case for AI-era science: machines now produce results that humans have not yet verified, and the institutions of peer review must catch up. Whatever the review concludes, the review itself — led by the conjecture’s own authors — is how the field will decide.
Related: OpenAI Tightens ChatGPT Weapons Limits for Teens in Violence-Prevention Push and Anthropic Files S-1 for Blockbuster IPO, to List on Nasdaq Under Ticker ANT.
FAQ
What is the exponential time hypothesis? — The claim that 3-SAT cannot be solved in subexponential time. Does this solve P vs NP? — No: the bound is subexponential, not polynomial. Has the disproof been verified? — Not yet: an independent review led by Impagliazzo, Paturi and Williams is under way.
Sources
- AFP via Malay Mail, 31 Oct 2026 — “I’d rather shake a human hand: Scientist’s bittersweet reaction as AI threatens to overturn his life’s work”
- TechTimes, 27 Oct 2026 — “Two AI Systems Disproved Same Complexity Conjecture Independently on Same Day”
- OfficeChai — “Anthropic Says Claude Too Independently Disproved Exponential Time Hypothesis”
- Anthropic preprint “A 2^o(n) Time Algorithm for 3-SAT” (Hexagon, 27 Oct 2026)
